Gauge Fixing and Renormalization in Topological Quantum Field Theory*

SLAC-PUB-4630 May 1988 P3

ROGER BROOKS, DAVID MONTANO, AND JACOB SONNENSCHEIN+

Stanford Linear Accelerator Center Stanford University, Stanford, California 94 309

ABSTRACT

In this letter we show how Witten’s topological Yang-Mills and gravitational quantum field theories may be obtained by a straightforward BRST gauge fixing procedure. We investigate some aspects of the renormalization of the topological Yang-Mills theory. It is found that the beta function for the Yang-Mills coupling constant is not zero.

Submitted to Phys. Lett.

* Work supported by the Department of Energy, contract DE-AC03-76SF00515. + Work supported by Dr. Chaim Weismann Postdoctoral Fellowship. 1. INTRODUCTION

Recently, there has been an effort to study the topology of four dimensional manifolds using the methods of quantum field theory!‘-‘] This was motivated by Donaldson’s use of self-dual Yang-Mills equations to investigate the topology of low dimensional manifolds!” The work of Floer on three manifolds, as interpreted by Atiyah!“can be understood as a modified non-relativistic field theory. Atiyah then conjectured that a relativistic quantum field theory could be used to study Donaldson’s invariants on four manifolds. This led Witten to work out a series of preprints in which just such topological quantum field theories (TQFT) were described.14-61 In the first paper a TQFT was proposed whose correlation func- tions were purely topological and reproduced Donaldson’s invariants!Q’ A later paper extended these ideas to a more complicated Lagrangian which could be used to study similar invariants in topological gravity!51These TQFT’s all possess a fermionic symmetry analogous to a BRST symmetry.

In this letter we will show how both of these theories may be obtained through a very simple gauge fixing procedure. A prescription for how quantum field theory calculations may be done for the TQFT will be presented. These calculations will allow us to study the perturbative renormalizability of the theory. It will be shown that the beta function for the coupling constant is not zero. This suggests that there is a quantum contribution to the stress-energy tensor which may indicate a metric dependence of the partition function.

Throughout this work we will use the notations of refs. [4] and [5]. However, in our discussion of the Yang-Mills TQFT, actions will be defined in terms of Lagrangians via I = s d4x ,/$3-C, where g is the determinant of the metric. 2. WITTEN’S TOPOLOGICAL YANG-MILLS THEORY

We begin by deriving Witten’s Lagrangian for the non-abelian Yang-Mills (YM) theory’*‘by gauge fixing the local transformation

&A,” = 8,” . (2.1)

Here, Aaa is a gauge field in the adjoint representation (index a). The gauge fixing will be done using the following principles as guidelines: First, the BRST gauge fixing procedure will be employed in order that we manifestly maintain the fermionic symmetries of ref. [4]. Second, because of their importance in the latter work, we will seek to preserve the scaling and U symmetries. Finally, we would like a YM invariant action at the end of the process. *Explicit coupling constant (ee) dependence will be given in our expressions. For this we define the YM covariant derivative as D, G d, + eo[Aa, 1.

We have found that the gauge fixing ansatz which leads to Witten’s theory is Fap + tap E 0, where k,p = &QI+F~~ is the dual of the YM field strength, F Q. To see this, let us assume that there is some Lagrangian, &, which is both YM and 61 invariant. Two examples are fZo = 0 and fZo oc [F,pbfl]. Both of these are topological with the Euclidean integral of the second yielding the Pontryagin index (winding number). The former is trivially invariant while the latter is invariant under ;I transformations if: (i) the Bianchi Identity for the YM field-strength, D,i@ = 0, is used, and (ii) the parameter 8, asymptotically drops off as one power faster than the gauge field. This is the requirement that 8, does not change the Pontryagin number. Given, lo, we BRST gauge fix by writing down the gauge fixing and Faddeev-Popov Lagrangians:

where we have introduced the BRST operator, iI, through the definition 61 z

3 i&l, for some constant anti-commuting parameter, E. Traces in the adjoint representation are implied in all of our Lagrangians. This particular Lagrangian maintains the BRST symmetry:

A 61&z" =hxa ,

&baa = 0 , (2.3) iIx aPa -B= ‘-@“, &B al3a -0- .

The anti-ghost field, x”p, and the BRST auxiliary field, Bap are anti-symmetric and self-dual. The field, $a, is the ghost conjugate to x@.

A few remarks about eqns. (2.2) and (2.3) are in order. To obtain Witten’s theory, we must pick the gauge oe = 0 for the gauge parameter. The operator 81 yields the usual off-shell, nilpotent, BRST algebra. The equation of motion of this auxiliary field yields the transformation law for x, as given in ref. [4]. After integrating out B, eqn. (2.3) is only an on-shell symmetry, as to be expected. The equation of motion which must be used to maintain this symmetry is obtained from the anti-ghost variation: Dra$~l + ~,p,sD7+~ = 0. Now, without this restriction on $, there are four gauge parameters. However, x only contains three “degrees of freedom”. This suggests that there is further gauge fixing to be done which will reduce the freedom of the ghost field. There must be an additional symmetry in the theory. As we will soon see, this symmetry forces the introduction of two additional commuting fields, 4 and X, and another anti- commuting field, r]. As a result, we will have the full multiplet of fields as given in ref. [4]. We now turn our attention to finding this symmetry while keeping our off-shell BRST invariance.

We do not have to look far, as the ghost Lagrangian above has a ghostly symmetry. It is invariant under the following transformations:

&aa = i(Da4)a ,

apa - &B - - ie0[4, xap]” ,

4 where 4” is a secondary ghost (or G-ghost) field and 8~ is the BRST operator of the ghost symmetry. To gauge fix this G-symmetry, we begin by writing

G?+FP= &~[ico~(Da$~ + sb) + clxaPBaB] , P-5) where & E 81 + &, the c; are arbitrary real constants, X is the G-anti-ghost, s is the G-gauge parameter and b is the G-auxiliary field. However, there are two more symmetries we must worry about. They are the scaling and U symmetries ‘*I. The global scaling and U symmetry weights of the fields (A, 4, X, ~,LJ,x) are (l,O, 2,1,2) and (0,2, -2,l, -l), respectively. In order to maintain these two invariances (in particular, the latter one), we must take b E eo[qS,q], for some anti-commuting field, 7, which is the transform of X; i.e., &X E 2~. This latter field has scaling and U weights 2 and -1, respectively. Observe that &r] # 0, contrary to what one would expect from traditional BRST gauge fixing. Short of introducing new fields, we find that the ansatz used above is unique. As the algebra of the symmetry, given in eqn. (2.4), c 1oses only up to an ordinary YM gauge transformation [*I, we have 8~7 = -eo$[4, X]. The natural choices for the constants ci are co = -i and cl = 5. This is the case since on manifolds of the form M = Y3 x R1, these choices maintain the time-reversal symmetry of the Lagrangian. Putting all of the above together, we find

L: - LO = (f?) + dG))G$‘+Fj= = -ix”‘Da$~p - iqDa$a

+ iADa-Da4 - i$oX[lliarOa] - ieo4[xaP, xap]

+ =0(+[rl, rll + $0[A Xl”)

+ iB”8sap + ~B*‘(Fa~ + Pap) (2.6)

= f(F + k)” - ix”‘Dapclp - iqDatia

+ iXD”Dad - iecJ[tia,$,] - $xd[xaP,xap]

+ se0W[w] + $0[4, Xl") ,

5 where the second equation follows from the first after use of the auxiliary field’s equation of motion. The Lagrangian (2.6) can be brought to the form C - LO = --$l’, where fZ’ is independent of the coupling constant, by a resealing of the fields. It should be observed that for reasons of renormalizability we cannot choose s = 0. Once again naturalness arguments favor the choice s = -$, which maintains the time-reversal symmetry of the Lagrangian. We have thus obtained Witten’s full YM Lagrangian (see eqns. (2.13) and (2.41) in ref. [4]) by straightforward gauge fixing.

3. RENORMALIZATION IN WITTEN’S YM TQFT*

Practical quantum field theory computations using the Lagrangian given in eqn. (2.6) become manageable if we make use of the following identification which is suggested by the time-reversal symmetry of the Lagrangian:

1 (3.1)

Under time-reversal we have : 4 + A, x -+4, (34 $” -+xa, xa + -fj”.

This symmetry suggests a “particle-antiparticle” relationship between 4 and A, and $J” and xa, respectively. This becomes even more suggestive when we rewrite the Lagrangian using the above identification:

,f - lo =i(F + k)2 - ix.J?)F$ + iXD”D& P-3) -

* In this section, we will only consider a flat Euclidean background.

6 where @F = I’PD,, with

rO = uO ~ uO, rl = o3 @ ia2, r2 = iU2 @CT', IIT3 = O1 @ ia ' (3.4

Here the @ are the standard Pauli matrices. We then have that the I’ll matrices obey the following algebra: {ri,rj} = - 2@, [ri,rj] = +krk, P-5) [r”,ri] = 0.

Hence, the I’p satisfy a U(2) algebra. It should also be noted that it is not possible to factor out a matrix (e.g. 7’) from the above in such a way that the ?j‘ = r”I’p obey the Clifford algebra. Instead, we have I’pfv + ]TprV = 2gpY, where l? = (I’O, -Pi).

Some simple calculations show that this theory is so constructed that it is renormalizable with only one coupling constant. Indeed, it is manifest from the derivation of the previous section that the time-reversal and BRST symmetries enforce the single coupling constant renormalization. It was suggested in ref [4] that the Lagrangian of eqn. (3.2) h as an unconventional N = 2 supersymmetry. The latter theory also has a single coupling constant. We see that the [4,X12 term is required as mentioned earlier, since counter-terms of such a form will be necessary to renormalize the box diagrams arising from the 4x2 and X+2 couplings.

We will now compute the coupling constant renormalization of this theory. Before proceeding we will need to gauge fix the Yang-Mills symmetry. In the absence of the 61 symmetry, this would be done in the standard way by introducing the anti-commuting ghosts c and E. However, we must be careful. For our calculations we will choose the covariant gauge, d,Ap = 0, and the Feynman-‘t Hooft gauge for the gauge boson propagator. Now, because of the gauge field’s

7 transformation law, given in eqn. (2.3), we will generate an additional ghost term of the form: ~a,$~. Notice that this is purely kinetic, so it will not contribute to our p-function. The Feynman rules for this theory are then given in Table 1. Through straightforward calculations we can now verify that the coupling constant is renormalized as follows:

2e3C2(G) + qe5). PC’) = - (41r)2 where C2 (G) is the second Casimir in the adjoint representation of the group G.

For any computation of /3( e) we will need to know the field strength renormalization. The only new diagrams arising in this computation are given in Table 2. One immediately observes that the + - x contribute like fermions in the adjoint representation of the gauge group; while the 4 - X give the standard scalar contribution. Adding the pure Yang-Mills diagrams we get that there is no field strength renormalization in the Feynman-‘t Hooft gauge. Now to fin- ish this calculation we just borrow the standard results from, for example, the boson-ghost-ghost vertex. It should be noted that the value for p(e) given above agrees with those in N = 2 super-Yang-Mills theory!‘]

The nonvanishing of p(e) may have some implications. The most significant will be that the energy-momentum tensor may not be a BRST commutator. Thus, we would expect that the theory may possess a dependence on the local properties of the background metric after all. In this case, the trace of the energy-momentum tensor will be proportional to the p-function. We did not find a way to generate a Ff counter-term in perturbation theory. Should it not be possible to generate such a counter-term, we must begin with Lo having a separate coupling constant for Fk, ie.

Lo = eFf . P-7) 4. WITTEN’S TOPOLOGICAL GRAVITY

We proceed now to derive Witten’s topological gravity action [S] following the same YM fixing procedure as we used above for the topological gauge field theory. Here the fundamental transformation law on the tetrad is

AA = eBl!3e bIea a AB,/i& - (4-l)

We introduce the BRST gauge fixing Lagrangian and its associated action.

L(I) = L$i+FG = ijI[xABCDwABCD], I(‘) = d4x det e - l(I), (4.2) / where det e is the determinant of the tetrad. WABCD is the self-dual part of the Weyl tensor and transforms as spin (0,2) of Sum x SU,(2). XABCD is an anti-ghost with the same spin. The transformations of xABCD and WABCD are given by

&ABCD =BABCD

~IWABCD =:[($J,&~‘&DA~ - eEA$,gDaDp@ABi’) (4.3) + (5 permutations of A, B, C, D)] ,

where BABCD is an auxiliary commuting ghost field and the variation of WABCD, which is given in ref. [S], follows from the variation: 81etA = eezB+AB,Ag. As in the topological gauge theory, the ghost Lagrangian

L(I) = iB ABCDW~~~~ - iXABCDiIWABCD , (4.4

is invariant under the following local symmetry:

~G@AB,A~ = - i(ezAD,CBg i- eEADaCAg •k e:gD,C,A -k eEgD,CAA) ,

iG~ABCD =A EBCD[(egkDaCAk + eaAkD,CEk + (cyclic perm.)] 4%

-- i ABCD a D cXk + iDa(ea . cXiXABCD)] . 2% exk a xx P-5)

These transformation laws are covariant with respect to SU,c(2) x SUEZ(~) trans-

9 formations, global U symmetry and scale invariance. Whereas &$,~,~fi has the same stucture as 8&a (in the topological YM theory with CQ = e&Cx’ being the analog of +), 6A~ B ABCD is proportional to the covariant derivative of Cxl;-, unlike iGBap. The source of this difference is that gGFPv has only one covariant derivative of $J whereas eqn. (4.3) includes D,D& terms. In any event the proof that L(I) is invariant under (4.5) proceeds as follows. First we write

iGLc(I)= iiGBABCDwABCD- ixABCDiGi~wABcD, (4.6) since &xABCD = &WABCD = 0. Hence

&(‘) = iWABcD [ aXEBCD [ (f&DacAA + eaARDaCE,) + (Cyclic perm.)]

ABCDea + ;x xkDaC xk + (DaWABCD)(e~~CXkXABCD)] -ix ABCD8~i~W~~~~. (4.7) Now by substituting

&~~WABCD=~[(~G$JAB~ ‘R~~A~- e~~ef,gDaDp&+ABiir) + (5 permutations of A, B, C, D)] i = + qwE,,D [(e&D,CAk + eaAkDaC,k) + (cyclic perm.)]

-- a w~BcDd&DacXk - iCaD,wABcD , 2 (4.8) which was given in ref. [S], it is clear that &t(I) = 0. Continuing to follow the procedure of section 2 we now add to L(I) a BRST gauge fixing term of the following form:

L(G) = LL?,,, = &[@eaAA(DaBBB)$AB ~g + $ABCDBA~CD] (4-g) ,

The Q field has spin (0,O) and conformal weight 2. It is introduced here following the same reasoning as in ref. [S], namely so that B,A has conformal weight 1.

10 This allows the first term in eqn. (4.9) to transform with a definite weight under local conformal transformations. Note that this term, which in our procedure follows naturally from the fixing of the ghost symmetry, is exactly the term that allowed Witten to introduce kinetic terms (denoted as Z in ref. [S]) for C and some of the components of $J. Now using the equation of motion for BABCD we finally get

L = ;WABcDWABCD - iXABCD($‘ABjBRcD~,j- e~Ae~,DaDpbABi’)

EBCD a D,cA~ + eaA'DaCEk) + ~XABCDX ceEk

ABCD @XT?ff CxiD,x~~c~)

+L3 , (4.10) where Ls (eqn(28) in [S]) is given by

L3 = @[A BBeaAADav,hAB ~g

-- fp PAADpBBB)(eziD,CBg + eiAD,CAg + e:,jDaCBA + eg@aCAA)

- iBBB,a . $AC~Ad-Da$AB Ag + iBCde‘&+AB~ArjDa$AB ~g cc 2 ,

+ ~DaBB~(eaB~~AcA~~Ac~~ + eaELi?,bABAkq!)A/') +. . .] . (4.11) where ‘. . .’ denotes other @ dependent terms. In analogy with the term cos&.v[Xb] in eqn. (2.5) we can add to eqn. (4.9) a term of the form &[@caDaBAAXAA] which will generate the analog of the last line in eqn. (2.6).

11 5. c ONCLUSIONS

We have shown how to derive Witten’s topological YM and gravitational theories by standard gauge fixing. This was done in a two-step process. First, the YM field-strength is gauge fixed to be anti-self-dual (an anti-instanton, for example). The resulting Lagrangian was then found to be invariant under a symmetry on the BRST ghost and auxiliary fields. Choosing a gauge slice for this last symmetry while maintaining the global scaling and ghost number (U) symmetries, led us to the full YM-TQFT. Following this procedure, we were also able to derive the gravitational TQFT.

One-loop calculations in the YM theory indicate that the coupling constant p-function does not vanish. Indeed, we find that it is the same as that of the N = 2 pure super-YM theory. This non-zero result means that the bare coupling constant in front of the F2 term in the Lagrangian, is divergent. As Fj also appears with the same coupling constant, we must find a counter-term to remove this infinity. However, although a counter-term of the form F2 is generated at the one-loop level, we were unable to find a Fi term. In this case we would have to start off with fZe = OF@ in order to maintain the renormalizability of the theory. Considering these results, it would be interesting to investigate the metric dependence of the theory.

Acknowledgements: It is our pleasure to thank M. Peskin for numerous useful discussions and for reading the manuscript.

Note added: Upon completion of this work we received two preprints related to Witten’s TQFT. Is1

12 TABLE 1: Feynman Rules

x$ Propagator: r$X Propagator:

Ax+ Vertex:

A& Vertex: -iefabc(r + g)p

&+!qb Vertex:

r-Traces: TABLE 2: One-loop contributions to p(e)

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